How do I find the maximum or minimum of a quadratic?

The max/min is at the vertex: h = -b/(2a), k = f(h). If a > 0, k is minimum. If a < 0, k is maximum.

Can I solve quadratics without the formula?

Yes. Factoring, completing the square, and inspection using sum/product of roots are alternative methods.

Math

Quadratic Formula Calculator

Solve any quadratic equation ax² + bx + c = 0 with step-by-step solutions. See discriminant, roots, vertex form, and a graph of the parabola.

Quick Answer

The quadratic formula solves ax² + bx + c = 0: x = (-b ± √(b² - 4ac)) / 2a. The discriminant b² - 4ac determines the nature of roots: positive = two real roots, zero = one repeated root, negative = two complex roots.

Enter Coefficients

For the equation ax² + bx + c = 0, enter values for a, b, and c. (a cannot be zero)

x² - 5x + 6 = 0

a (coefficient of x²)

b (coefficient of x)

c (constant)

Solution

x₁ (First Root)

x₂ (Second Root)

Discriminant

Nature of Roots

Two Real Distinct

Vertex (h, k)

(2.5, -0.25)

Axis of Symmetry

x = 2.5

Y-Intercept

Opens

Upward

Graph

VertexRoot(s)

Step-by-Step Solution

Step 1: Identify coefficients

a = 1, b = -5, c = 6

Step 2: Calculate the discriminant

D = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1

Step 3: Apply the quadratic formula

x = (-(-5) ± √(1)) / (2 × 1)

x = (5 ± 1) / 2

Step 4: Solve

x₁ = 3

x₂ = 2

Step 5: Vertex form

h = -b/(2a) = -(-5)/(2×1) = 2.5

k = f(h) = 1(2.5)² + (-5)(2.5) + 6 = -0.25

Vertex form: y = (x - 2.5)² + -0.25

Additional Properties

Sum of roots: 5 (-b/a)

Product of roots: 6 (c/a)

About This Tool

The Quadratic Formula Calculator solves any quadratic equation of the form ax² + bx + c = 0 using the quadratic formula. It provides both roots, the discriminant, vertex coordinates, a step-by-step solution, and a visual graph of the parabola. This tool is essential for algebra students, engineers, and anyone working with polynomial equations.

The Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / 2a works for every quadratic equation, regardless of whether the roots are rational, irrational, or complex. The ± symbol means you calculate twice: once with addition and once with subtraction, giving you both solutions. This formula was known to ancient Babylonian, Indian, and Persian mathematicians, though the modern notation developed in the 17th century.

Understanding the Discriminant

The discriminant D = b² - 4ac determines the nature of the solutions without actually solving the equation. If D is greater than 0, you get two distinct real roots and the parabola crosses the x-axis at two points. If D equals 0, you get one repeated root (the parabola touches the x-axis at its vertex). If D is less than 0, there are no real roots; instead you get two complex conjugate roots, and the parabola does not cross the x-axis.

Vertex Form and Graphing

Every quadratic can be written in vertex form: y = a(x - h)² + k, where (h, k) is the vertex of the parabola. The vertex is the minimum point when a is greater than 0 (parabola opens upward) or the maximum when a is less than 0 (parabola opens downward). The axis of symmetry is the vertical line x = h. Converting to vertex form by completing the square is a fundamental algebraic technique.

Real-World Applications

Quadratic equations model projectile motion, optimization problems, revenue/cost analysis, area calculations, and many physical phenomena. When you throw a ball, its height over time follows a quadratic equation. When a business models profit as a function of price, the optimal price is at the vertex. Physics, engineering, economics, and computer graphics all rely heavily on quadratic equations.

Frequently Asked Questions

What does it mean when the discriminant is negative?

A negative discriminant means the quadratic equation has no real number solutions. Instead, it has two complex conjugate roots involving the imaginary unit i (where i = the square root of -1). Graphically, the parabola does not cross or touch the x-axis. Complex roots always come in conjugate pairs: if one root is a + bi, the other is a - bi.

Can 'a' be zero in the quadratic formula?

No. If a = 0, the equation becomes bx + c = 0, which is a linear equation with one solution x = -c/b. The quadratic formula requires a non-zero value for 'a' because division by 2a would be undefined, and the equation would no longer be quadratic (having a degree of 2).

How do I find the maximum or minimum value of a quadratic?

The maximum or minimum occurs at the vertex. The x-coordinate of the vertex is h = -b/(2a), and the y-coordinate (the max or min value) is k = f(h). If a > 0, the parabola opens upward, so k is the minimum. If a < 0, the parabola opens downward, so k is the maximum. This is fundamental to optimization problems in algebra and calculus.

What are complex roots and when do they occur?

Complex roots occur when the discriminant (b^2 - 4ac) is negative. They take the form a + bi and a - bi, where i represents the square root of -1. Complex roots are not visible on a standard x-y graph because they don't correspond to real x-intercepts. They are essential in electrical engineering, signal processing, and quantum mechanics.

Is there a method to solve quadratics without the formula?

Yes. You can factor the equation if the roots are rational (e.g., x^2 - 5x + 6 = (x-2)(x-3) = 0). You can complete the square, which is how the quadratic formula itself is derived. For simple cases, you can also use the relationship that the sum of roots = -b/a and product of roots = c/a to find solutions by inspection.

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